In oil and gas exploration, rock formations or other subsurface media are often characterized based on seismic surveys, i.e., seismic measurements in conjunction with computational modeling of the media and simulation of seismic wave propagation from the seismic sources to the seismic receivers. For example, in a typical land-acquisition geometry, one or more artificial seismic sources, such as explosives, truck-mounted vibrators, or hammers, may be deployed on the earth's surface to excite seismic waves in the subsurface rock formation, and a plurality of seismic receivers, such as geophones, may be distributed on the surface around the source(s) to measure seismic waves resulting from the excitation (such as reflections off geological boundaries). In order to derive information about the formation (such as the thicknesses and materials of its various layers, and the location of oil or gas reservoirs therein) from the seismic measurements, the measurements may be compared with the results of a computational simulation that is based on a computational model reflecting assumptions about the subsurface formation. Discrepancies between measurements and simulation suggest inaccuracies in the model.
Rock formations often include multiple horizontal layers of different materials. When acting as propagation media for seismic waves, such formations display vertical transverse isotropy, i.e., their material properties are independent of the direction of wave propagation within a horizontal plane (i.e., a plane perpendicular to the vertical symmetry axis of isotropy). Such vertically transverse isotropic (VTI) media can be adequately modeled with existing finite-difference elastic modeling approaches, which generally involve numerically solving elastic wave equations discretized over a suitable three-dimensional volume of the formation, using a suitable discretization grid. For tilted transversely isotropic (TTI) media, where a geological boundary and, thus, the plane of isotropy is angled relative to the horizontal plane, however, the conventional approaches generally do not perform satisfactorily in that—in order to avoid inaccurate results or simulation artifacts—they use so fine a grid spacing that they can quickly become computationally intractable.